Electrons held by magnetic field

1. The problem statement, all variables and given/known data
Electrons are held by a magnetic field B in circular orbit in a vacuum chamber. The electrons are acclerated by increasing the magnetic flux linking the orbit. Prove that the average magnetic field over the plane of the orbit must be twice the magnetic field at the orbit if the orbit is to remain fixed as the electron's energy is increased.

I haven't been able to figure this out but I think I have the right idea.

As the magnetic field increases, it induces an electric field which does work on the electron, increasing its velocity. Since the magnetic field at the orbit is increasing, the radius stays constant since

[itex]r = mv/eB[/itex].

Appying Faraday's law, I find that after an infinitesimal period of time, the velocity increases by dv = (e/m)(r/2) dB.